Spin-accumulation can be produced at the edges of a conducting bar at zero external magnetic field, by injecting an electric current into a non-magnetic material with high spin-orbit coupling [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. This effect is called Spin Hall Effect (SHE). It has been predicted some decades ago [7]}, and described in the framework of various theoretical models [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}. However, a description that takes into account the non-equilibrium nature of the electric screening occurring in the SHE is still an open problem. One of the main difficulty is the same as for the classical Hall-effect [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}: it is due to the fact that the values of the charge accumulation at the edges of the Hall bar are not directly imposed by the external constraints. Instead, for the stationary state, the accumulation of electric charges and spins at the edges is generated by the system itself, in reaction to the action of the internal magnetic field (the spin-orbit effective magnetic field), together with to stationary current injected from the electric generator, and for a given geometry. As a consequence, the determination of a unique solution to the drift-diffusion equations at stationary state based on the local boundary conditions is problematic [11]}.
